(a) How do you find vertical asymptotes of rational functions? (b) Let s be the rational function $$s ( x ) = \frac { a _ { n } x ^ { n } + a _ { n -

Reggie

Reggie

Answered question

2021-05-04

(a) How do you find vertical asymptotes of rational functions? (b) Let s be the rational function s(x)=anxn+an1xn1++a1x+a0bmxm+bm1xm1++b1x+b0$$
How do you find the horizontal asymptote of s? (c) Find the vertical and horizontal asymptotes of f(x)=5x2+3x24

Answer & Explanation

Brighton

Brighton

Skilled2021-05-05Added 103 answers

a) We find a vertical asymptote of a rational function such that we find a rational zeros of the denominator i.e the line x = a is a, vertical asymptote of a rational function, where a is zero of the denominator.
b) Let s(x) be a rational function
s(x)=anxn+an1xn1++a1x+a0bmxm+bm1xm1++b1x+b0
then
1. If n < m, then s(x) has horizontal asymptote y = 0,
2. lf n = m, then s has horizontal asymptote y=anbm
3. If n > m, then s has no horizontal asymptote.
c) We want to find the denominator in the factored form
f(x)=5x2+3x24
Sine, we can express the denominator in the factored form
x24=(x2)(x+2)
we can see that zeros 2 and -2, therefore from the part a) the lines x=2 and x=-2 are vertical asymptotes.
We can see that n=2 and m=2 i.e n=m. So, from the part b) we obtain that the horizontal asymptote is y=5.

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